It means that choice of z(x,y) satisfies the equation. I.e plug in z(x,y) into LHS and into the RHS and they should be equal.chemphys1 said:
The chain rule is a mathematical rule used to find the derivative of a composite function. In other words, it helps us determine how a small change in one variable will affect the overall function. In multiple variable calculus, the chain rule is essential because it allows us to find the partial derivatives of a function with respect to each variable. This is crucial in many scientific fields, such as physics and engineering, where functions often depend on multiple variables.
In multiple variable calculus, the chain rule is applied by taking the partial derivative of the outer function with respect to each variable and then multiplying it by the partial derivative of the inner function with respect to the same variable. This process is repeated for each variable in the function. In other words, we use the chain rule to find the rate of change of a function with respect to each variable separately.
Let's say we have a function f(x,y) = x^2 + y^3. If we wanted to find the partial derivative of this function with respect to x, we would first take the derivative of the outer function, which is 2x, and then multiply it by the derivative of the inner function, which is 1. This gives us a partial derivative of 2x. We would repeat this process for the partial derivative with respect to y, giving us a final result of 2x + 3y^2.
The chain rule and the product rule are closely related. In fact, the chain rule is often used in the product rule when differentiating a product of functions in multiple variables. The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second plus the second function times the derivative of the first. When dealing with functions in multiple variables, we use the chain rule to find the derivatives of each individual function in the product.
The chain rule is essential in understanding how a small change in one variable can affect the overall function. In scientific fields, we often deal with functions that depend on multiple variables, and using the chain rule allows us to analyze how each variable contributes to the function. It also helps us find the critical points and extrema of a function, which are crucial in many scientific applications, such as optimization and modeling.
